3D F–K and finite-difference dip moveout in cross-spreads

A 3D F–K dip-moveout (DMO) is developed, which is applicable to data acquired in an elementary single-fold cross-spread. The key idea is that a 3D log-stretch transform and the inherent regularity of the cross-spread geometry make it possible to transform 3D Fourier DMO. The derived theory generalizes the 2D Fourier shot-gather DMO in the log-stretch domain; 2D turns out to be a special case. Similarly to 2D, the cross-spread DMO becomes convolutional after multidimensional logarithmic stretch. The proposed method works for orthogonal and slanted acquisition geometries; the cross-spread DMO relationships are found to be independent of the intersection angle of the shot and receiver lines. In contrast to integral (Kirchhoff-style) methods, the cross-spread F–K DMO does not degrade from the inevitable irregularity in 3D sampling of offsets in a CMP gather. The newly derived F–K DMO operator can be approximated by finite-difference (FD) schemes; the low-order FD cross-spread DMO equation is shown to be the 3D extension of the Bolondi and Rocca offset continuation. It is shown that F–K and low-order FD operators are effective in a synthetic case.